Nonlinear Sloshing in Fixed and Vertically Excited Containers (original) (raw)
Related papers
Journal of Fluids and Structures, 2003
Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model simulates two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference timestepping scheme on adaptively mapped grids. A s-transformation in the vertical direction that stretches directly between the free surface and bed boundary is applied to map the moving free-surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells.
Nonlinear Free-Surface and Viscous-Internal Sloshing in 2-D Numerical Wave Tanks
Volume 3: Materials Technology; Ocean Engineering; Polar and Arctic Sciences and Technology; Workshops, 2003
This paper examines free-surface and internal-pycnocline sloshing motions in 2-D numerical wave tanks subjected to horizontal base excitation. In all of the cases studied, the rectangular tank of liquid has a width-to-depth ratio of 2. The first set of results are based on an inviscid, fully nonlinear finite difference free-surface model. The model equations are mapped from the physical domain onto a rectangular domain. Case studies at and off resonance are presented illustrating when linear theory is inadequate. The next set of results are concerned with analyzing internal waves induced by sloshing a density-stratified liquid. Nonlinear, viscous flow equations are solved. The influence of the side-wall boundary layers on sloshing motions as well as the onset of internal breaking of the primary sloshing mode are discussed. The frequencies that characterize the motion of internal waves are also reported.
Classification of three-dimensional nonlinear sloshing in a square-base tank with finite depth
Journal of Fluids and Structures, 2005
The paper classifies steady state three-dimensional resonant waves in a square-base tank by using the asymptotic modal system proposed by the authors in 2003. The effective frequency domains of stable steady state motions are analysed versus mean fluid depths and forcing amplitude. The results are validated by experiments both qualitatively and quantitatively. r (O.M. Faltinsen). related to this study are reviewed by and . Three different approaches to theoretical sloshing modelling are distinguished. One of them focuses on low-order asymptotic mathematical theories and appropriate Hamiltonian formalism for the system of ordinary differential equations governing the dominating standing waves. Another approach is based on computational fluid dynamics (CFD) [see surveys by , Gerrits (2001), Celebi and Akyildiz ]. The third approach deals with multimodal/pseudospectral methods. Such methods are able to provide in different versions both analytical and numerical studies and 'build a bridge' between the first and second ones. All three approaches have their advantages and disadvantages from mathematical, physical and engineering points of view outlined in details by the mentioned surveys. Links, common features and differences should be demonstrated. This is a difficult problem with regard to the lower-order mathematical theories (first approach) and modal/pseudospectral methods [see some details given by ; Hill ]. Both approaches reduce the original free boundary problem to systems of ordinary differential equations with finite nonlinear kernel and often focus on nonlinear steady state waves. The difference is that the modal methods account for the full set of activated modes and their arbitrary initial perturbation, while the first approach studies the behaviour of the leading modes. Generally speaking, the multidimensional modal approach is more general, because under some additional asymptotic assumptions a corresponding low-order Hamiltonian system can be derived from the modal systems. The opposite is not true. AlthoughHill presented a version of single-dominant theory of two-dimensional sloshing, where the behaviour of some higher modes can be restored, his scheme is invalid for arbitrary initial conditions (for the higher modes) and requires the single harmonic forcing in a very small vicinity of the primary resonance.
Sloshing effects in periodically and seismically excited tanks
Proceedings of the 5th Intl. World Congress on Computational Mechanics
A fully nonlinear finite difference model has been developed based on inviscid flow equations. Numerical experiments of sloshing wave motion are undertaken in a 2-D tank which is moved both horizontally and vertically. Results of liquid sloshing induced by harmonic and earthquake base excitations are presented for small to steep non-breaking waves for steepness up to 0.3. Good agreement for small horizontal forcing amplitude is achieved between the numerical model and first order small perturbation theory. For large horizontal forcing, nonlinear effects are captured by the numerical model. The effect of the simultaneous vertical and horizontal excitation in comparison with the pure horizontal motion is examined. It is shown that vertical excitation causes the instability of the combined motion for a certain set of frequencies and amplitudes of the vertical motion. It is also found that in addition to the resonant frequency of the pure horizontal excitation, two additional resonance frequencies exist due to the combined motion of the tank. The dependence of the nonlinear behaviour of the solution on the wave steepness is discussed. It is found that for the present problem nonlinear effects become important when the steepness reaches about 0.1.
Sloshing motions in excited tanks
Journal of Computational Physics, 2004
A fully non-linear finite difference model has been developed based on inviscid flow equations. Numerical experiments of sloshing wave motion are undertaken in a 2-D tank which is moved both horizontally and vertically. Results of liquid sloshing induced by harmonic base excitations are presented for small to steep non-breaking waves. The simulations are limited to a single water depth above the critical depth corresponding to a tank aspect ratio of h s =b ¼ 0:5. The numerical model is valid for any water depth except for small depth when viscous effects would become important. Solutions are limited to steep non-overturning waves. Good agreement for small horizontal forcing amplitude is achieved between the numerical model and second order small perturbation theory. For large horizontal forcing, nonlinear effects are captured by the third-order single modal solution and the fully non-linear numerical model. The agreement is in general good, both amplitude and phase. As expected, the third-order compared to the second-order solution is more accurate. This is especially true for resonance, high forcing frequency and mode interaction cases. However, it was found that multimodal approximate forms should be used for the cases in which detuning effects occur due to mode interaction. We present some test cases where detuning effects are evident both for single dominant modes and mode interaction cases. Furthermore, for very steep waves, just before the waves overturn, and for large forcing frequency, a discrepancy in amplitude and phase occurs between the approximate forms and the numerical model. The effects of the simultaneous vertical and horizontal excitations in comparison with the pure horizontal motion and pure vertical motion is examined. It is shown that vertical excitation causes the instability associated with parametric resonance of the combined motion for a certain set of frequencies and amplitudes of the vertical motion while the horizontal motion is related to classical resonance. It is also found that, in addition to the resonant frequency of the pure horizontal excitation, an infinite number of additional resonance frequencies exist due to the combined motion of the tank. The dependence of the non-linear behaviour of the solution on the wave steepness is discussed. It is found that for the present problem, non-linear effects become important when the steepness reaches about 0.1, in agreement with the physical experiments of Abramson [Rep. SP 106, NASA, 1966].
The semi-Lagrangian procedure is widely used for updating the fully-nonlinear free surface in the time domain. However, this procedure is only available to cases when the body surface is vertical near the waterline. Present study introduces an improved semi-Lagrangian procedure which removes this ‘vertical-wall’ limitation. Coupling with the boundary element method, the improved semi-Lagrangian procedure is applied to the simulation of fully-nonlinear sloshing waves in non-wall-sided tanks. From the result comparison with the open source CFD software OpenFOAM, it is confirmed that this numerical scheme could guarantee a sufficient accuracy. Further series studies on 2D and 3D fully-nonlinear sloshing waves in wedged tanks are performed. Featured phenomena are observed which are distinct from those in wall-sided tanks. Application of an improved semi-Lagrangian procedure to fully-nonlinear simulation of sloshing in non-wall-sided tanks. Available from: https://www.researchgate.net/publication/274193712\_Application\_of\_an\_improved\_semi-Lagrangian\_procedure\_to\_fully-nonlinear\_simulation\_of\_sloshing\_in\_non-wall-sided\_tanks [accessed May 14, 2015].
Nonlinear Liquid Sloshing in Rectangular Tank
2007
Nonlinear analytical solution of liquid sloshing behavior in rigid rectangular tank is presented here. Theory of perturbation with the concept of velocity potential is applied to formulate the analytical solutions. The analytical procedure transforms the nonlinear problem into a multi-stage linear problem in such a way that the solution is efficiently and easily obtained in consecutive manner. Closed form solutions for velocity potential function, free surface shape, sloshing force and wave breaking are formulated for a sinusoidal horizontal excitation along the length of the tank. Significant participation of the second sloshing mode is observed in addition to the first sloshing mode when the excitation frequency is near to the first sloshing mode or when the excitation amplitude is relatively high. It results significant deviation between upward and downward movements of free surface. The solution also indicates the phenomenon of super-harmonic resonance. Extensive shaking table e...
Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth
Journal of Fluid Mechanics, 2000
The discrete infinite-dimensional modal system describing nonlinear sloshing of an incompressible fluid with irrotational flow partially occupying a tank performing an arbitrary three-dimensional motion is derived in general form. The tank has vertical walls near the free surface and overturning waves are excluded. The derivation is based on the Bateman-Luke variational principle. The free surface motion and velocity potential are expanded in generalized Fourier series. The derived infinite-dimensional modal system couples generalized time-dependent coordinates of free surface elevation and the velocity potential. The procedure is not restricted by any order of smallness. The general multidimensional structure of the equations is approximated to analyse sloshing in a rectangular tank with finite water depth. The amplitude-frequency response is consistent with the fifth-order steady-state solutions by . The theory is validated by new experimental results. It is shown that transients and associated nonlinear beating are important. An initial variation of excitation periods is more important than initial conditions. The theory is invalid when either the water depth is small or water impacts heavily on the tank ceiling. Alternative expressions for hydrodynamic loads are presented. The procedure facilitates simulations of a coupled vehicle-fluid system.
Nonlinear Free-Surface and Viscous-Internal Sloshing
Journal of Offshore Mechanics and Arctic Engineering, 2005
This paper examines free-surface and internal-pycnocline sloshing motions in twodimensional numerical wave tanks subjected to horizontal excitation. In all of the cases studied, the rectangular tank of liquid has a width-to-depth ratio of 2. The first set of results are based on an inviscid, fully nonlinear finite difference free-surface model. The model equations are mapped from the physical domain onto a rectangular domain. Case studies at and off resonance are presented illustrating when linear theory is inadequate. The next set of results are concerned with analyzing internal waves induced by sloshing a density-stratified liquid. Nonlinear, viscous flow equations are solved. Two types of breaking are discussed. One is associated with a shear instability which causes overturning on the lee side of a wave that moves towards the center of the container; this wave is generated as the dominant sloshing mode recedes from the sidewall towards the end of the first sloshing cycle. The other is associated with the growth of a convective instability that initiates the formation of a lip of heavier fluid above lighter fluid behind the crest of the primary wave as it moves up the sidewall. The lip grows into a bore-like structure as it plunges downward. It falls downward behind the primary wave as the primary wave moves up the sidewall and ahead of the primary wave as this wave recedes from the sidewall. This breaking event occurs near the end of the first cycle of sloshing, which is initiated from a state of rest by sinusoidal forcing.